Archived web site of the Workshop on the Atlas of Lie Groups and Representations, Salt Lake City, July 10–21, 2017. All Atlas workshops
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jda@Leonidas:~$ cd atlasSoftware/master/atlasofliegroups/atlas-scripts/
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ ../atlas all
This is 'atlas' (version 1.0.7, axis language version 0.9.6),
the Atlas of Lie Groups and Representations interpreter,
compiled on Jul 17 2017 at 20:01:50. http://www.liegroups.org/
atlas> <twisted_dual.at
Starting to read from file 'twisted_dual.at'.
Added definition [3] of dual_block: ([Param],mat->[Param],[int])
Added definition [4] of dual_block: (Param,mat->[Param],[int])
Added definition [2] of KL_Q_polynomials_via_dual: ([Param],mat->[[vec]])
Defined signs: (vec->[[vec]])
Redefined KL_P_polynomials: ([Param],mat->[[vec]])
Completely read file 'twisted_dual.at'.
atlas> <sl3
Starting to read from file 'sl3.at'.
Variable G: RealForm
Variable delta: mat
Variable p: Param
Variable B: [Param]
Variable P: [[vec]]
Variable Pt: [[vec]]
Variable Pbig: [[vec]]
P:
+1 -1 -1 +1 0 0
0 +1 0 -1 -1 0
0 0 +1 -1 0 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
Pt:
+1 +1
0 +1
Pbig:
+1 0 -1 +1 0 0
0 +1 -1 0 +1 0
0 0 +1 -1 -1 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
|B|=6
delta-fixed parameters:=[0,3]
complete indices=[(0,1),(0,-1),(1,0),(3,1),(3,-1),(4,0)]
delta_action=[0,2,1,3,5,4]
Completely read file 'sl3.at'.
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atlas> quit
Bye.
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ evince ~/Desktop/hermitianFormsSMFrefB.pdf
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$
jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ ../atlas all
This is 'atlas' (version 1.0.7, axis language version 0.9.6),
the Atlas of Lie Groups and Representations interpreter,
compiled on Jul 17 2017 at 20:01:50. http://www.liegroups.org/
atlas>
atlas>
atlas>
atlas>
atlas>
atlas> <twisted_dual.at
Starting to read from file 'twisted_dual.at'.
Added definition [3] of dual_block: ([Param],mat->[Param],[int])
Added definition [4] of dual_block: (Param,mat->[Param],[int])
Added definition [2] of KL_Q_polynomials_via_dual: ([Param],mat->[[vec]])
Defined signs: (vec->[[vec]])
Redefined KL_P_polynomials: ([Param],mat->[[vec]])
Completely read file 'twisted_dual.at'.
atlas> set G=SL(3,R)
Variable G: RealForm
atlas>
atlas> set delta=distinguished_involution (G)
Variable delta: mat
atlas> delta
Value:
| 1, 0 |
| 1, -1 |
atlas> simple_roots (G)
Value:
| 1, 1 |
| -1, 2 |
atlas> delta*simple_roots (G)
Value:
| 1, 1 |
| 2, -1 |
atlas>
atlas>
atlas> set p=trivial(G)
Variable p: Param
atlas> print_block(p)
Parameter defines element 3 of the following block:
0: 0 [C+,C+] 2 1 (*,*) (*,*) *(x=0,lam_rho= [0,0], nu= [0,0]/1) e
1: 1 [i2,C-] 1 0 (3,4) (*,*) *(x=1,lam_rho= [0,0], nu= [3,3]/2) 2xe
2: 1 [C-,i2] 0 2 (*,*) (3,5) *(x=2,lam_rho= [0,0], nu= [3,0]/2) 1xe
3: 2 [r2,r2] 4 5 (1,*) (2,*) *(x=3,lam_rho= [0,0], nu= [2,1]/1) 1^2xe
4: 2 [r2,rn] 3 4 (1,*) (*,*) *(x=3,lam_rho= [1,1], nu= [2,1]/1) 1^2xe
5: 2 [rn,r2] 5 3 (*,*) (2,*) *(x=3,lam_rho= [1,0], nu= [2,1]/1) 1^2xe
atlas> set Bfixed=fixed_block_of (delta,p)
Variable Bfixed: [Param]
atlas> #Bfixed
Value: 2
atlas> for p in Bfixed do prints(p)
G >
G > od
final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1)
Value: [(),()]
atlas>
atlas>
atlas> set B=block_of (trivial(G))
Variable B: [Param]
atlas> set P=KL_P_signed_polynomials(B)
Variable P: [[vec]]
atlas> P
Value: [[[ 1 ],[ -1 ],[ -1 ],[ 1 ],[ ],[ ]],[[ ],[ 1 ],[ ],[ -1 ],[ -1 ],[ ]],[[ ],[ ],[ 1 ],[ -1 ],[ ],[ -1 ]],[[ ],[ ],[ ],[ 1 ],[ ],[ ]],[[ ],[ ],[ ],[ ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ 1 ]]]
atlas> printPolyMatrix (P)
+1 -1 -1 +1 0 0
0 +1 0 -1 -1 0
0 0 +1 -1 0 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas>
atlas>
atlas> set Q=KL_Q_polynomials(B)
Variable Q: [[vec]]
atlas> printPolyMatrix (Q)
+1 +1 +1 +1 +1 +1
0 +1 0 +1 +1 0
0 0 +1 +1 0 +1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas> printPolyMatrix (P*Q)
+1 0 0 0 0 0
0 +1 0 0 0 0
0 0 +1 0 0 0
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas>
atlas>
atlas> print_extended_
print_extended_block print_extended_composition_series
print_extended_character_formula print_extended_indices
atlas> print_extended_indices (B,delta)
|B|=6
delta-fixed parameters:=[0,3]
complete indices=[(0,1),(0,-1),(1,0),(3,1),(3,-1),(4,0)]
delta_action=[0,2,1,3,5,4]
atlas>
atlas>
atlas>
atlas>
atlas>
atlas> set Pbig=big_KL_P_signed_polynomials(B,delta)
Variable Pbig: [[vec]]
atlas> printPolyMatrix (Pbig)
+1 0 -1 +1 0 0
0 +1 -1 0 +1 0
0 0 +1 -1 -1 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas> printPolyMatrix (P)
+1 -1 -1 +1 0 0
0 +1 0 -1 -1 0
0 0 +1 -1 0 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas> printPolyMatrix (Pbig)
+1 0 -1 +1 0 0
0 +1 -1 0 +1 0
0 0 +1 -1 -1 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas> Bf
Error during analysis of expression at <standard input>:48:0-2
Undefined identifier 'Bf'
Expression analysis failed
atlas> Bfixed
Value: [final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1),final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1)]
atlas> set Pdelta=KL_P_signed_polynomials(delta,Bfixed )
Error in expression KL_P_signed_polynomials(delta,Bfixed) at <standard input>:50:11-49
Failed to match 'KL_P_signed_polynomials' with argument type (mat,[Param])
Expression analysis failed
Command 'set Pdelta' not executed, nothing defined.
atlas> set Pdelta=KL_P_signed_polynomials(delta,p)
Error in expression KL_P_signed_polynomials(delta,p) at <standard input>:51:11-43
Failed to match 'KL_P_signed_polynomials' with argument type (mat,Param)
Expression analysis failed
Command 'set Pdelta' not executed, nothing defined.
atlas> whattype KL_P_signed_polynomials?
Overloaded instances of 'KL_P_signed_polynomials'
Param->[[vec]]
[Param]->[[vec]]
(Param,mat)->[[vec]]
([Param],mat)->[[vec]]
atlas> set Pdelta=KL_P_signed_polynomials(p,delta)
Variable Pdelta: [[vec]]
atlas> printPolyMatrix (Pdelta)
+1 +1
0 +1
atlas>
atlas>
atlas>
atlas>
atlas>
atlas> printPolyMatrix (P)
+1 -1 -1 +1 0 0
0 +1 0 -1 -1 0
0 0 +1 -1 0 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas> printPolyMatrix (Pdelta )
+1 +1
0 +1
atlas>
atlas>
atlas>
atlas>
atlas> printPolyMatrix (P)
+1 -1 -1 +1 0 0
0 +1 0 -1 -1 0
0 0 +1 -1 0 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas>
atlas> printPolyMatrix (Pbig )
+1 0 -1 +1 0 0
0 +1 -1 0 +1 0
0 0 +1 -1 -1 -1
0 0 0 +1 0 0
0 0 0 0 +1 0
0 0 0 0 0 +1
atlas>
atlas>
atlas> printPolyMatrix (Pdelta )
+1 +1
0 +1
atlas>
atlas>
atlas>
atlas> print_block(B)
Error in expression print_block(B) at <standard input>:77:0-14
Failed to match 'print_block' with argument type [Param]
Expression analysis failed
atlas> print_block(p)
Parameter defines element 3 of the following block:
0: 0 [C+,C+] 2 1 (*,*) (*,*) *(x=0,lam_rho= [0,0], nu= [0,0]/1) e
1: 1 [i2,C-] 1 0 (3,4) (*,*) *(x=1,lam_rho= [0,0], nu= [3,3]/2) 2xe
2: 1 [C-,i2] 0 2 (*,*) (3,5) *(x=2,lam_rho= [0,0], nu= [3,0]/2) 1xe
3: 2 [r2,r2] 4 5 (1,*) (2,*) *(x=3,lam_rho= [0,0], nu= [2,1]/1) 1^2xe
4: 2 [r2,rn] 3 4 (1,*) (*,*) *(x=3,lam_rho= [1,1], nu= [2,1]/1) 1^2xe
5: 2 [rn,r2] 5 3 (*,*) (2,*) *(x=3,lam_rho= [1,0], nu= [2,1]/1) 1^2xe
atlas>
atlas>
atlas>
atlas> print_extended_block (p,delta)
0: [3Ci ] +0; +3 x=0, [ 0, 0 ]
3: [3r ] +3; +0 x=3, [ 0, 0 ]
atlas>
atlas>
atlas>
atlas>
atlas> set E=E(delta,p)
Variable E: (InnerClass,mat,ratvec,vec,mat,ratvec,vec,mat,vec,vec)
atlas> display(E)
x=KGB element #3
gamma=[ 2, 1 ]/1
g=[ 1, 0 ]/1
lambda_rho=[ 0, 0 ]
tau=[ 0, 0 ]
l=[ 0, 0 ]
t=[ 0, 0 ]
atlas>
atlas>
atlas>
atlas>
atlas> parameter(E)
Value: final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1)
atlas> p
Value: final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1)
atlas> G:=SL(4,R)
Value: connected split real group with Lie algebra 'sl(4,R)'
atlas> print_ext_block (trivial(G),d)
Error during analysis of expression at <standard input>:96:0-30
Undefined identifier 'd'
Expression analysis failed
atlas> set delta=distinguished_involution (G)
Variable delta: mat (overriding previous instance, which had type mat)
atlas> print_ext_block (trivial(G),delta)
Error in expression print_ext_block(trivial(G),delta) at <standard input>:98:0-34
Failed to match 'print_ext_block' with argument type (Param,mat)
Expression analysis failed
atlas> print_ext_block (delta,trivial(G))
0 2Ci 1i1 2Ci: final parameter(x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1)
1 2Ci 1i1 2Ci: final parameter(x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1)
2 2C+ 1r1f 2C+: final parameter(x=2,lambda=[3,2,1]/1,nu=[0,1,-1]/2)
3 2Cr 1C+ 2Cr: final parameter(x=3,lambda=[3,2,1]/1,nu=[1,0,1]/1)
4 2Cr 1C+ 2Cr: final parameter(x=4,lambda=[3,2,1]/1,nu=[1,0,1]/1)
7 2i12 1C- 2i12: final parameter(x=7,lambda=[3,2,1]/1,nu=[2,2,0]/1)
8 2i12 1C- 2i12: final parameter(x=8,lambda=[3,2,1]/1,nu=[2,2,0]/1)
10 2C- 1i2f 2C-: final parameter(x=10,lambda=[3,2,1]/1,nu=[6,3,3]/2)
12 2r21 1r2 2r21: final parameter(x=12,lambda=[3,2,1]/1,nu=[3,2,1]/1)
13 2r21 1rn 2r21: final parameter(x=12,lambda=[4,3,1]/1,nu=[3,2,1]/1)
14 2rn 1r2 2rn: final parameter(x=12,lambda=[3,3,2]/1,nu=[3,2,1]/1)
atlas> print_extended_block (delta,trivial(G))
Error during analysis of expression at <standard input>:100:0-39
Type error:
Subexpression (delta,trivial(G)) at <standard input>:100:21-39
has wrong type: found (mat,Param) while (Param,mat) was needed.
Expression analysis failed
atlas> print_extended_block (trivial(G),delta)
0: [2Ci ,1i1 ] +0 +1; +3 +2 x= 0, [ 0, 0, 0 ]
1: [2Ci ,1i1 ] +1 +0; +4 +2 x= 1, [ 0, 0, 0 ]
2: [2C+ ,1r1f ] +10 +2; . +0,+1 x= 2, [ 0, 0, 0 ]
3: [2Cr ,1C+ ] +3 +7; +0 . x= 3, [ 0, 0, 0 ]
4: [2Cr ,1C+ ] +4 +8; +1 . x= 4, [ 0, 0, 0 ]
7: [2i12f,1C- ] +7 +3; +12,+13 . x= 7, [ 0, 0, 0 ]
8: [2i12f,1C- ] +8 +4; +12,-13 . x= 8, [ 0, 0, 0 ]
10: [2C- ,1i2f ] +2 +10; . +12,+14 x=10, [ 0, 0, 0 ]
12: [2r21f,1r2 ] +12 +14; +7,+8 +10 x=12, [ 0, 0, 0 ]
13: [2r21f,1rn ] +13 +13; +7,-8 . x=12, [ 1, 1, 0 ]
14: [2rn ,1r2 ] +14 +12; . +10 x=12, [ 0, 1, 1 ]
atlas>
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atlas>